# Onset of the wave turbulence description of the longtime behavior of the nonlinear Schr\"odinger equation

## TL;DR

This paper demonstrates that for the cubic nonlinear Schrödinger equation on a torus with random phase initial data, the evolution of Fourier coefficient magnitudes aligns with the wave kinetic equation over significant timescales, confirming predictions of wave turbulence theory.

## Contribution

It provides a rigorous link between the nonlinear Schrödinger equation and the wave kinetic equation for random initial data on a torus, validating wave turbulence predictions.

## Key findings

- Fourier coefficient moduli follow the wave kinetic equation on average
- Results hold for data with uniformly distributed independent phases
- Validates wave turbulence theory in a rigorous setting

## Abstract

Consider the cubic nonlinear Schr\"odinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.